Abstract

An investigation of the minimum number of intersecting beams that
is required for laser Doppler anemometry (LDA) incorporating only a
single detector is presented. We aim to provide decisive arguments
for using four beams as the minimum requirement for complete
three-dimensional velocity reconstruction even though three beams
supply three velocity components. We derive expressions for the
detected signals of the most general LDA system. From a matrix
analysis of these expressions, we conclude that there is no physically
realizable arrangement of three beams that results in complete
three-dimensional velocity reconstruction and that four beams is the
minimum number of beams required. We also determine the optimal
arrangement of the four incident beams for unambiguous LDA and for best
signal separation and immunity to minor optical alignment
errors. To ascertain the velocity components, we scan the specimen
in a precise manner relative to the point of focus of the beams,
whereas some other researchers alter the frequency of the incident
beams. The results obtained with these two methods are
equivalent. However, scanning is mechanically simpler than
frequency shifting and also allows for the formation of velocity
images—images of the flow velocity over a region in two- or
three-dimensional space. In particular, we examine systems that are
limited by the common practice of using only a single
high-numerical-aperture objective for both focusing and
detection. We show that using high-numerical-aperture objectives
results in the best signal differentiation and immunity to minor
alignment errors.

Figures (6)

Schematic diagram representing the photomultiplier signal
with (a) three incident beams and (b) four incident
beams. In both (a) and (b) there is a dc pedestal. It
is vital to separate the signals by as much as possible to avoid any
confusion in the detected beat frequencies.

Optical arrangement of the LDA system showing one (of
possibly many) incident beams i. Note that the use of
a single objective limits the possible incident-wave vectors because of
the NA of the element.

Shown is the collapse of a set of randomly chosen values
for the different parameters of the LDA system as they are altered by
iteration to a solution that ensures that all the beat frequencies are
uniformly separated.

Data from which the relations for α > π/2
were taken: Examination of many sets of variables that maintain the
maximum beat-frequency separation reveals the relations that hold
between different parameters. In particular, we see that
θw is always equal to θ4
(curve with squares), that θw always sums
with θ1 (curve with crosses) to equal π, and that
θw is independent of both ϕ1
(curve with circles) and ϕ4 (curve with
triangles).

Superposition of the Monte Carlo simulation results with
the theoretical results for the determinant of the arrangement matrix
[Eq. (15)] as a function of the half-angle of the NA. The
determinant is a measure of the optical arrangement—a zero determinant
means that no optical arrangement is possible. From this figure, we
can see that a fairly large NA is required for strong stability of the
optical LDA arrangement.